### Characterization of idempotent 2-copulas

#### Abstract

A 2-copula induces a transition probability function via

where , denoting the Lebesgue measurable subsets of. We say that a set is invariant under if for almost all , being the characteristic function of. The sets invariant under form a sub--algebra of theLebesgue measurable sets, which we denote . A set is called an atom if it has positive measure and if for any , is either or 0.A 2-copula is idempotent if . Here denotes the product defined in [1]. Idempotent 2-copulas are classified and characterized asfollows:

(i) An idempotent is said to be nonatomic if contains noatoms. If is a nonatomic idempotent, then it is the product of a leftinvertible copula and its transpose. That is, there exists a copula such that where

(ii) An idempotent is said to be totally atomic if there exist essentiallydisjoint atoms withIf is a totally atomic idempotent, then it is conjugate to an ordinal sumof copies of the product copula. That is, there exists a copula satisfying and a partition of such that

\begin{equation}F=C*(\oplus _{\cal P}F_k)*C^T \end{equation} where eachcomponent in the ordinal sum is the product copula .

(iii) An idempotent is said to be atomic (but not totally atomic) if contains atoms but the sum of the measures of a maximal collection ofessentially disjoint atoms is strictly less than 1. In this mixed case, thereexists a copula invertible with respect to and a partition of for which (1) holds, with being a nonatomic idempotent copula andwith for .

Some of the immediate consequences of this characterization are discussed.

DOI Code:
10.1285/i15900932v30n1p147

Keywords:

copula; idempotent; star product

copula; idempotent; star product

Classification:
60G07; 60J05; 60J25

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